The volume of a right circular cylinder is 770 cubic centimetres. If the circumference of the base is 22 cm, what is the curved (lateral) surface area of the cylinder in square centimetres? (Take π = 22/7)

Introduction / Context:
This problem involves a cylinder where volume and base circumference are given, and the curved surface area is required. Solving it requires the use of three key formulas: the circumference of a circle, the volume of a cylinder, and the curved surface area of a cylinder. It is an excellent test of the ability to combine multiple geometric relationships and perform algebraic manipulation to find both the radius and height of the cylinder.

Given Data / Assumptions:
- Volume of the cylinder V = 770 cubic centimetres.- Circumference of the base = 22 cm.- π is to be taken as 22/7.- Curved surface area (CSA) of a cylinder is CSA = 2 * π * r * h.- Volume of a cylinder is V = π * r^2 * h.

Concept / Approach:
First, we use the base circumference to find the radius r of the cylinder using the formula circumference = 2 * π * r. Once r is obtained, we substitute into the volume formula V = π * r^2 * h to solve for the height h. Finally, we use the formula for curved surface area (lateral surface area) CSA = 2 * π * r * h with the found values of r and h. The steps involve solving linear equations and careful substitution of numerical values for π, r, and h.

Step-by-Step Solution:
Step 1: Use the circumference formula: 2 * π * r = 22.Step 2: Substitute π = 22/7 into 2 * π * r = 22.Step 3: 2 * (22/7) * r = 22 ⇒ (44/7) * r = 22.Step 4: Solve for r: r = 22 * 7 / 44 = 7 / 2 = 3.5 cm.Step 5: Use the volume formula V = π * r^2 * h with V = 770.Step 6: r^2 = (3.5)^2 = 12.25.Step 7: Substitute: 770 = π * 12.25 * h.Step 8: Replace π with 22/7: 770 = (22/7) * 12.25 * h.Step 9: Compute (22/7) * 12.25 = (22/7) * (49/4) = (22 * 49) / 28 = (1078) / 28 = 38.5.Step 10: So 770 = 38.5 * h ⇒ h = 770 / 38.5 = 20 cm.Step 11: Now compute CSA = 2 * π * r * h.Step 12: CSA = 2 * (22/7) * 3.5 * 20.Step 13: 2 * (22/7) * 3.5 = 2 * (22/7) * (7/2) = 22.Step 14: Therefore, CSA = 22 * 20 = 440 square centimetres.

Verification / Alternative check:
An alternative verification is to compute the base area using r = 3.5 cm and h = 20 cm, then check if the volume matches 770 cubic centimetres. Base area = π * r^2 ≈ (22/7) * 12.25 = 38.5. Then Volume = base area * height = 38.5 * 20 = 770 cubic centimetres, which matches the given volume exactly. This confirms that r and h are correct and therefore the calculated curved surface area of 440 square centimetres is reliable.

Why Other Options Are Wrong:
- 880 sq cms: This is exactly double the correct answer and might result from mistakenly using 4 * π * r * h.- 220 sq cms: This is half of the correct answer, possibly from missing a factor of 2 in the CSA formula.- 660 sq cms: This is one and a half times the correct value and has no clear basis in the provided dimensions.- 330 sq cms: This does not correspond to any standard misapplication of formulas but is still inconsistent with the correct computations.

Common Pitfalls:
Students often forget to use the circumference to find the radius and instead guess the radius directly. Others might incorrectly substitute in the volume formula or forget to square the radius. Missing the factor of 2 in the curved surface area formula is another common error. A systematic approach, using each formula at the correct stage and checking intermediate results, helps avoid these mistakes.

Final Answer:
The curved surface area of the cylinder is 440 square centimetres.